The present invention relates to a new technique for eliminating overtone collisions in musical scales, and to a novel interval system for tuning of musical instruments wherein dissonances resulting from struck chords are eliminated. More particularly, the invention relates to an electronic musical instrument which reproduces musical scales so that normal struck chords, such as major fifths, do not have such dissonances. The invention is based on a wave system of communication which relies on a different basis of periodicity in wave propagation.
One problem in human communication since the first attempts to use and reproduce vibrations has been the development of a standard wave system of communication such that production of a wave of a given definition does not interfere with any other wave which occupies space. In the context of music, such interference causes overtone collisions, or dissonances, which can make individual notes, or melody lines in music difficult to identify and to follow.
Throughout history, there have been a number of interval system whose goal has been to minimize such dissonances. Four of the most widely-used such systems in auditory octave tuning for Western music are the Pythagorean, just intonation, meantone, and equal temperament intervals. These are defined, for example, in The New Harvard Dictionary of Music, (1986, Harvard Press). As described in that publication, pitch is calculated by a ratio involving logarithms, cents, and string lengths, all of which involve a measure of periodicity of waveforms based on .pi.=3.1416.
None of the above-mentioned interval systems suffices, by itself, to prevent overtone collisions. Various attempts have been made to combine various ones of these systems as appropriate in electronic musical instruments, to minimize the degree of overtone collision present. Some of these now will be discussed.
U.S. Pat. No. 4,152,964, which relates to a correction of the "larger tuning errors of equal temperament as each interval or chord is played", switches intervals between equal temperament and just intonation, depending on whether a chord or a single note is to be played. Also in that patent is a discussion of some of the inherent deficiencies in the mathematical base from which the various then-known intervals were derived. It is useful to consider that discussion here:
"No chromatic scale with tones of fixed pitch can yield perfectly tuned chords and also allow complete freedom of modulation. A scale composed of perfectly tuned chords must have notes whose frequencies form an arithmetical progression, while if the scale is to allow complete freedom of modulation, the notes must have frequencies that form a geometrical progression. In the first case, although the frequency differences are all congruent, the sizes of the various intervals, measured logarithmically, are not congruent with respect to the octave or with one another because the logarithms of simple interval ratios are irrational decimals. In the second case the sizes of the intervals, measured logarithmically, are congruent with one another and with the octave but now, since the interval ratios are all expressed as fractional powers of two, and hence irrational, all the intervals of such a scale except the octave are more or less out of tune. PA0 This dilemma which lies at the root of the difficulty of realizing just intonation with scales of fixed pitch, can be resolved by converting the present scale of equal temperament into a scale with tones of mutable pitch. Thus, the modulational advantage of the present scale is preserved by retaining the tempered fourths and fifths without alteration while the harmonic potentialities are greatly enlarged by the use of a keyboard controlled computer which automatically shifts the pitch of certain notes to correct the larger tuning errors of the scale. PA0 Generally, an electronic musical instrument is tuned in an equally tempered scale so that it is easy to modulate or transpose to other keys or to make ensemble performance with other musical instruments. However, when the electronic musical instrument is thus tuned with the equally tempered scale, such chord tones as major triad chord tones are not produced in perfect consonant intervals so that it constitutes one of the factors that disturb harmony. For example, when major triad chord tones are produced by a just intonation scale, the frequency ratio of the root note tone to the major third note tone is just "4:5", and the frequency ratio of the root tone to the perfect fifth note tone is just "2:3" and accordingly "4:6". On the other hand, when the major triad chord tones are produced with the equally tempered scale, the frequency ratio of the root note to the major third note is "4:5.03984". Thus, the pitch of the major note in the equally tempered scale become higher by 14 cents than that of the major third note in the just intonation scale. Furthermore, when major triad chord tones are produced in an equally tempered scale, the frequency ratio of the root note to the perfect fifth note is "4:5.993228". Thus, the pitch of the perfect fifth note in the equally tempered scale is lower by 2 cents than that of the perfect fifth note, in a just intonation scale. As a consequence, where chord tones are produced in a just intonation scale, clear tones can be produced with consonant intervals. On the other hand, where chord tones are produced in an equally tempered scale, the tones become a bit unharmonic. PA0 "On the other hand, an instrument tuned according to the equal-temperament cannot obtain perfect chords when compared to an instrument tuned according to the temperament of just intonation. However, the instrument tuned according to the temperament of equal-temperament is capable of obtaining chords which sound substantially natural, and in addition, the modulation operation is simple. For this reason, general electronic keyboard instruments, piano, and the like were conventionally tuned according to the temperament of equal-temperament. However, the chords obtained from the keyboard instruments which are tuned according to the temperament of equal-temperament are not perfect chords as described before, and these instruments are unfit for use in teaching during chorus practice, for example. PA0 The disadvantages of fixed scale systems will be evident from the following description: It is well known that the just scale CDE.sub.1 FGA.sub.1 G.sub.1 C which is generated by the perfectly tuned chords FA.sub.1 C, CE.sub.1 G and GB.sub.1 D, contains the imperfect minor chord DFA.sub.1 in which the note D is a comma too sharp relative to the note A.sub.1. On a fixed scale basis, a perfectly tuned chord D.sub.1 FA.sub.1 can be had only as the submediant triad in the key of F Major or as the mediant triad in the key of B Flat Major, by momentarily turning on either of these tonality stops. A further disadvantage of just intonation on a fixed scale basis is that the same mis-tuned triad DFA which would also be contained in the dominant ninth chord GB.sub.1 DFA.sub.1, renders that chord even more dissonant than the same chord in equal temperament. PA0 For centuries numerous scholars and critical listeners have argued that the influences of fixed-pitch instruments have contributed to a loss of correct pitch and have caused vocalists and instrumentalists not constrained by fixed pitch to sing and play "out of tune" either for equally tempered or "just" performance. Basic to this problem has been the lack of technological development in instruments for either tempered tuning or just intonation. An examination of the abundant literature on the subject discloses that no fixed-pitch or keyboard instruments have previously been proposed or built capable of approaching precisely equal tempered intervals, nor any that could accurately produce just intonation and all of its enharmonic notes or modulational pitch changes for either instructional or performance use.
The technique of the above-discussed patent, then, does not represent a systematic approach to elimination of overtone collision. The lack of a systematic approach makes the scales reproduced by the electronic musical instrument difficult to transpose into different keys. While equal temperament facilitates such transposition, the problem of overtone collision is serious.
Another illuminating discussion is found in U.S. Pat. No. 4,248,119, relative to the incompatibility of freedom of modulation in one system with chords of the same system, wherein it is stated:
Thus, in both of the just-quoted patents, there is recognition that no single interval system has been able to provide sufficient harmony for the different situations in which both single notes and chords are struck.
In U.S. Pat. No. 4,498,363, it is stated:
In the above-mentioned U.S. Pat. No. 4,152,964, there is discussion of some of the disadvantages of both the just intonation and the equal temperament scales.
Thus, there has been clear recognition in the prior art that no fixed-scale system has been known which eliminates overtone collision by itself. An interesting summary of the problem is provided in U.S. Pat. No. 4,434,696, wherein it is stated: